Supervisors info:
Απόστολος Γιαννόπουλος, Καθηγητής, Τμήματος Μαθηματικών Ε.Κ.Π.Α.
Γατζούρας Δημήτριος, Καθηγητής Τμήματος. Μαθηματικών Ε.Κ.Π.Α.
Χελιώτης Δημήτριος, Αναπληρωτής Καθηγητής Τμήματος Μαθηματικών Ε.Κ.Π.Α.
Summary:
We define the coarse Ricci curvature of a metric space, equiped with a Markov chain. The idea behind this definition is based on the following property of a Riemann manifold with positive Ricci curvature: small balls are closer in Wasserstein distance than their centers. On the one hand, this definition is compatible with Riemann manifolds, in which case the coarse Ricci curvature is the same with the Ricci curvature, after a rescaling. On a general setting, it implies a great range of results, such as a spectral gap bound, a Gaussian-type concetration of measure theorem, a version of a logarothmic Sobolev inequality and an exponential concetration theorem.
